Inverse Trig Functions and Solving Trig Equations

Why inverses need restricted domains

Sine fails the horizontal line test, so it has no inverse — until we restrict it. The AP conventions are fixed:

• $\arcsin x$: outputs in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ (quadrants IV and I),
• $\arccos x$: outputs in $[0,\pi]$ (quadrants I and II),
• $\arctan x$: outputs in $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.

Consequence: $\arcsin(\sin x)=x$ only when $x$ already lies in the principal range. Outside it, the calculator quietly returns a different, co-terminal-flavored angle.

The solving workflow

1. Isolate one trig function of one argument.
2. Find the reference solution with an inverse function (or a known special angle).
3. Use unit-circle symmetry to find the second solution in one period: $\sin$ pairs $\theta$ with $\pi-\theta$; $\cos$ pairs $\theta$ with $-\theta$ (i.e. $2\pi-\theta$); $\tan$ needs no second family — its period is already $\pi$.
4. Append $+2\pi k$ (or $+\pi k$ for tangent) for the general solution.
5. If the argument is $bx$, solve for $bx$ first over an interval $b$ times as long, then divide — a compressed equation has $b$ times as many solutions per window.